Thursday, June 9, 2011

My infinity can beat up your infinity

Indeed, many 19th century mathematicians found infinity to be vaguely distasteful, and they felt it had no place in serious mathematical discussion. At best, infinity was something for philosophers to discuss, and you can imagine the sort of disdain with which such pronouncements were made. It was in that context that Georg Cantor published the first proof of the existence of infinity in 1874.

Born in Russia but raised in Germany, Cantor provided a stunning and instantly controversial proof that not only defined the nature of infinity, but it also revealed that multiple infinities existed, and some were larger than others. What made his achievement all the more remarkable was that he had built the entire thing out of an ancient and seemingly useless branch of mathematics known as set theory. Basically, it was the mathematical equivalent of building an interstellar drive out of a wheelbarrow. ...

The classical example is a set containing the natural numbers, which are all the non-negative integers beginning with zero. ... [Its] cardinality is actually aleph-null, or aleph-zero, which is the smallest type of infinity. ...

If we want to get to aleph-one, the next order of infinity, we'll need to come up with something that is uncountably infinite. ... The most famous example is the set of all real numbers, which includes all the natural numbers, all the rational numbers, all the irrational numbers such as the square root of 2, and the transcendental numbers such as the values pi or e. ...

Can we go still further to aleph-two, aleph-three, and so on and so forth? It is indeed possible to take things further, and all we need is one more concept: power sets.

A power set for any number N is the set of all the subsets of set N. ...

[If] we take the set of all real numbers - or aleph-one - then the power set of aleph-one will have a greater cardinality, which means it must at least be aleph-two. We can keep doing this forever, with the power set of aleph-two giving us aleph-three, the power set of aleph-three giving us aleph-four, and so on.

And here's the really weird part. Since you can repeat the power set operation an infinite number of times, it stands to reason that there must eventually be an aleph-infinity...or, perhaps more accurately, and aleph-aleph-null. And even that might still pale in comparison to Georg Cantor's notion of an absolute infinite that transcended all attempts to express infinity within set theory. For his part, Cantor suspected that the absolute infinite was God.
--Alasdair Wilkins, Gizmodo, on infinity upon infinity

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